![Figure 3: A, Steady state solution of the diffusion-reaction model of species body mass evolution (eq. [4]), with , , and bm p 0 M p 2 gmin chosen to minimize the tail-weighted Kolmogorov-Smirnov (wKS) distance between the model and the empirical distribution, with (smoothed) empirical data for 8,617 avian species. B, The plane of wKS dis-(m, b) tances (lighter values correspond to smaller wKS), showing a systematic relationship between m and b for producing realistic (small wKS) avian mass distributions. The form of this relationship is similar to that of mammals (fig. 2B), suggesting a similar overall macroevolutionary mechanism. A color version of this figure is available in the online edition of the American Naturalist.](/figures/figure-3-a-steady-state-solution-of-the-diffusion-reaction-3f60cbc7.webp)
![Figure 2: A, Solution of the diffusion-reaction model of species body mass evolution (eq. [4]) with (smoothed) empirical data for 4,002 terrestrial mammals from the late Quaternary. Parameters m and xmin were estimated from empirical data on extinct terrestrial mammals since the Cretaceous-Paleogene boundary, while b was chosen to minimize the tail-weighted Kolmogorov-Smirnov (wKS) distance between the model and the empirical distribution. B, The plane of wKS distance of(m, b) the specified model and the empirical data for terrestrial mammals (lighter values correspond to smaller wKS). The distinctive groove (line) demonstrates that a systematic relationship between the bias parameter m and the extinction parameter b is necessary to produce realistic mass distributions (small wKS). The parameter pair used in A is marked. Dashes demarcate the line of no within-lineage bias ( ). A colorm p 0 version of this figure is available in the online edition of the American Naturalist.](/figures/figure-2-a-solution-of-the-diffusion-reaction-model-of-gcmbzfp6.webp)
vol. 173, no. 2 the american naturalist february 2009
How Many Species Have Mass M?
Aaron Clauset,
1,
*
David J. Schwab,
2
and Sidney Redner
1,3
1. Santa Fe Institute, Santa Fe, New Mexico 87501; 2. Department of Physics and Astronomy, University of California, Los Angeles,
California 90024; 3. Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215
Submitted June 16, 2008; Accepted August 25, 2008; Electronically published December 17, 2008
Online enhancements: color versions of figures.
abstract: Within large taxonomic assemblages, the number of spe-
cies with adult body mass M is characterized by a broad but asym-
metric distribution, with the largest mass being orders of magnitude
larger than the typical mass. This canonical shape can be explained
by cladogenetic diffusion that is bounded below by a hard limit on
viable species mass and above by extinction risks that increase weakly
with mass. Here we introduce and analytically solve a simplified
cladogenetic diffusion model. When appropriately parameterized, the
diffusion-reaction equation predicts mass distributions that are in
good agreement with data on 4,002 terrestrial mammals from the
late Quaternary and 8,617 extant bird species. Under this model, we
show that a specific trade-off between the strength of within-lineage
drift toward larger masses (Cope’s rule) and the increased risk of
extinction from increased mass is necessary to produce realistic mass
distributions for both taxa. We then make several predictions about
the evolution of avian species masses.
Keywords: macroevolution, species body mass distribution, diffusion,
Cope’s rule, mammals, birds.
Introduction
For a large taxonomic group under stable macroevolu-
tionary conditions, how many species have a mass M? This
question has wide implications for the evolution and dis-
tribution of many other species characteristics that cor-
relate strongly with body mass, including life span, life
history, habitat, metabolism, and extinction risk (Calder
1984; Brown 1995; Bennett and Owens 1997; Cardillo et
al. 2005).
Extant species, including mammals, birds, fish, and in-
sects, exhibit a canonical form of the species mass distri-
bution (Stanley 1973; Kozłowski and Gawelczyk 2002; Al-
len et al. 2006), in which the typical mass is an intermediate
value; for example, in mammals, the typical mass is that
of the common Pacific rat (Rattus exulans; 40 g). Larger
or smaller species, in turn, are significantly less common
* Corresponding author; e-mail: aaronc@santafe.edu.
Am. Nat. 2009. Vol. 173, pp. 256–263. 䉷 2009 by The University of Chicago.
0003-0147/2009/17302-50539$15.00. All rights reserved.
DOI: 10.1086/595760
but asymmetrically so; the largest species, such as the ex-
tinct imperial mammoth (Mammuthus imperator;10
7
g)
for terrestrial mammals, are many orders of magnitude
larger, while the smallest species are only a little smaller,
for example, the Remy’s pygmy shrew (Suncus remyi; 1.8
g).
The ubiquity of this distribution of species masses (fig.
1A) suggests the existence of a universal evolutionary
mechanism. A theoretical explanation of this distribution
may shed light both on the interaction between ecological
and macroevolutionary processes (Stanley 1975) and on
long-term trends in species mass (Alroy 2000a, 2000b),
including Cope’s rule, the oft-studied notion that species
mass tends to increase within a lineage over evolutionary
time (Stanley 1973; Alroy 1998). Clauset and Erwin (2008)
recently showed, by comparing extensive computer sim-
ulations with empirical data, that cladogenetic diffusion
in the presence of a taxon-specific lower limit on mass
M
min
and extinction risks that grow weakly with mass can
explain both the canonical form described above and the
precise form of the distribution of terrestrial mammal
masses.
Here, we present a simplified three-parameter version
of the Clauset and Erwin (CE) model for the species mass
distribution and solve this model in the steady state. Com-
paring the predictions of this simplified model with species
mass data for the same 4,002 terrestrial mammals from
the late Quaternary (Smith et al. 2003, 2007; henceforth
denoted “Recent” mammals, which includes species that
have become extinct during the Holocene), we reproduce
Clauset and Erwin’s results. We then show that the model’s
predictions, when appropriately parameterized, are also in
good agreement with data for 8,617 extant avian species
(Dunning 2007).
The Clauset-Erwin Model
Many models of the variation of species body mass over
evolutionary time assume a cladogenetic diffusion process
(Stanley 1973; McKinney 1990; McShea 1994; Kozłowski
and Gawelczyk 2002), where each descendant species’ mass
How Many Species Have Mass M? 257
Figure 1: A, Illustration of the similarity of the body mass distributions
for different taxonomic groups, using data for 4,002 Recent terrestrial
mammals (Smith et al. 2003, 2007) and 8,617 extant birds (Dunning
2007). For visual clarity, the empirical distributions have been smoothed
using a Gaussian kernel (Wasserman 2006) and rescaled to lay on the
same abscissa interval. B, Schematic showing the basic cladogenetic dif-
fusion process by which species mass varies over evolutionary time. The
mass of a descendant M
D
is related to the mass of its ancestor M
A
by a
random multiplicative factor l, which represents the total selective in-
fluence on species mass from all sources; that is, . A colorM p lM
DA
version of this figure is available in the online edition of the American
Naturalist.
M
D
is related to its ancestor’s mass M
A
by a random mul-
tiplicative factor l; that is, (fig. 1B), where lM p lM
DA
represents the total selective influence on the descendant
species’ mass from all sources. Clauset and Erwin (2008)
studied a family of such diffusion models, whose form and
parameters could be estimated empirically. Under their
model, the evolution of species mass is bounded on the
lower end by hard physiological limits, perhaps from met-
abolic (Pearson 1948, 1950; West et al. 2002) or morpho-
logical (Stanley 1973) constraints, and on the upper end
by an extinction risk that increases weakly with mass (Liow
et al. 2008). Near the lower limit, however, within-lineage
changes to body mass become increasingly biased toward
larger masses; that is, Aln lS increases as .M r M
min
Using model parameters estimated from ancestor-
descendant data on extinct North American terrestrial
mammals since the Cretaceous-Paleogene boundary (Al-
roy 2008), Clauset and Erwin simulated 60 million years
of mammalian body mass evolution and found that the
predicted distribution closely matches that of 4,002 Recent
terrestrial mammal species. They also found that simpler
diffusion models—for example, those that omitted the
lower limit, an increased extinction risk from increased
mass, or the increased bias toward larger masses as M r
—predicted significantly different distributions.M
min
Thus, these three mechanisms can explain not only the
canonical form of the species mass distribution but also
its taxon-specific shape.
A Mathematically Solvable Version
The CE model, however, remains too complex for math-
ematical analysis, even though it omits many ecological
and microevolutionary processes such as interspecific
competition, predation, and population dynamics. By sim-
plifying its assumptions slightly, we can formulate a dif-
fusion-reaction model of species body mass evolution,
which retains most of the key features of the CE model
and which can be mathematically analyzed. This model
can then be used to make inferences about body mass
evolution without resorting to laborious simulations.
Let denote the number of species with logarithmicf(x, t)
mass at time t; that is, we transform the CEx p lnM
model’s multiplicative diffusion process into an additive
diffusion process on a logarithmic scale. The cladogenesis
mechanism leads to two offspring species in which x r
. In the continuum limit, a value (Cope’sx ⫹ lnl Aln lS
1 0
rule) corresponds to positive drift velocity in the equation
of motion for . In the CE model for terrestrial mam-f(x, t)
mals, the drift velocity was estimated from fossil data to
increase like M
1/4
near the lower limit on species mass, a
feature that Clauset and Erwin (2008) found was necessary
to accurately predict the number of small-bodied mam-
mals. (Whether this quarter-power form is related to the
quarter-power scaling commonly found elsewhere in the
body size literature [Savage et al. 2004] remains to be seen.)
For mathematical simplicity, however, we will ignore this
dependence at the expense of possibly misestimating the
number of small-bodied species. We also assume that se-
lection pressures on body mass are roughly independent,
implying that the distribution of changes l is approxi-
mately lognormal (but see Clauset and Erwin 2008).
258 The American Naturalist
The feature that two offspring are produced at each
update step corresponds to a population growth term in
the equation of motion that is proportional to f itself. The
extinction probability may also be represented by a loss
term that is proportional to f. For terrestrial mammals,
recent empirical studies (Liow et al. 2008) support the
assumption that the probability per unit time of a species
becoming extinct p
e
(x) grows weakly with its mass. The
CE model uses a simple parameterization of this behavior:
, where , which corresponds to anp (x) p A ⫹ Bx B ≥ 0
e
extinction probability that grows logarithmically with spe-
cies mass.
Combining these three elements—diffusion, cladogen-
esis, and extinction—we may write the continuum equa-
tion of motion for the number of species with massf(x, t)
at time t asx p ln M
2
⭸f ⭸f ⭸ f
⫹ v p D ⫹ (1 ⫺ A ⫺ Bx)f,(1)
2
⭸t ⭸x ⭸x
where is the drift velocity (strength of Cope’s
v p Aln lS
rule) and the variance is the diffusion coef-
2
D p Aln lS
ficient. (In physics, eq. [1] is called the convection-
diffusion equation or the Fokker-Planck equation.) This
equation, however, omits the lower limit on species body
mass, which we incorporate momentarily.
The time-dependent equation of motion itself may be
useful for studying evolutionary trends in species body
mass or for making inferences about correlated extinction
or speciation events. For our purposes, however, we are
mainly interested in its stationary solution. Such a steady
state should exist whenever all species within the taxon
experience roughly the same set of macroevolutionary se-
lective pressures, that is, under stable macroevolutionary
conditions. To derive this solution, we set the time deriv-
ative in equation (1) to zero to obtain
f ⫺ mf ⫹ (a ⫺
, where , , , andbx)f p 0 m p
v/D a p (1 ⫺ A)/D b p B/D
the prime denotes differentiation with respect to x.We
now eliminate the first derivative term by introducing
to transform the steady state equation to
mx/2
f p e w
2
m
w ⫹ a ⫺⫺bx w p 0.
[( ) ]
4
This equation can be brought into the form of the standard
Airy’s differential equation (Abramowitz and Stegun
1972):
w ⫺ zw p 0, (2)
where we introduce the new variable
1/3
z p b x ⫺
and the prime now denotes differentiation
⫺2/3 2
b (a ⫺ m /4)
with respect to z. The general solution to equation (2) is
, where Ai(z) and Bi(z) are thew(z) p c Ai(z) ⫹ c Bi(z)
12
Airy functions. Since there can be no species with infinite
mass, we may set . The parameter c
1
is then deter-c p 0
2
mined by the normalization of f(x).
Thus, we can now write the species mass distribution
as
2
m
mx/2 1/3 ⫺2/3
f(x) ∝ eAib x ⫺ ba⫺ .(3)
[()]
4
Including now the taxon-specific lower limit x
min
on spe-
cies mass implies the constraint for the steadyf(x ) p 0
min
state solution and allows us to eliminate one parameter
from equation (3). Using the fact that the first zero of the
Airy function is located at , which we nowz p ⫺2.3381 …
0
require to coincide with , gives the constraintx p x
min
2
m
1/3 ⫺2/3
z p b x ⫺ ba⫺ ,
0min
()
4
which we may solve for a. Inserting this result into equa-
tion (3) yields
mx/2 1/3
f(x) ∝ eAi[b (x ⫺ x ) ⫹ z ](4)
min 0
as the steady state solution for the species mass distribution.
If the lower limit x
min
is known, this simple-minded
model has only two parameters: m, associated with the
biased diffusion process, and b, associated with the ex-
tinction process. A positive bias in the diffusion m
1 0
(Cope’s rule) has several systematic effects on the distri-
bution of species masses: it (1) pushes the left tail of the
distribution away from the lower limit at x
min
, (2) shifts
the modal mass toward slightly larger values, and (3) ex-
tends the right tail of the distribution. In contrast, in-
creasing b implies that species become extinct with greater
probability for a given mass M, which contracts the right
tail of the distribution. For a given bias m and number of
species n, the parameter b also sets an effective upper limit
on the expected maximum observed mass within the taxon
without invoking a hard boundary, for example, from bio-
mechanical constraints (McMahon 1973).
Mammalian Body Mass Evolution
We now test the predictions of this simple mathematical
model using empirical data for 4,002 Recent terrestrial
mammals (Smith et al. 2003, 2007) and 8,617 birds (Dun-
ning 2007). In the former case, we take , theM p 1.8 g
min
size of the smallest known mammal, and we estimate
and (SE) from Al-
v p 0.109 Ⳳ 0.021 D p 0.508 Ⳳ 0.027
How Many Species Have Mass M? 259
Figure 2: A, Solution of the diffusion-reaction model of species body
mass evolution (eq. [4]) with (smoothed) empirical data for 4,002 ter-
restrial mammals from the late Quaternary. Parameters m and x
min
were
estimated from empirical data on extinct terrestrial mammals since the
Cretaceous-Paleogene boundary, while b was chosen to minimize the
tail-weighted Kolmogorov-Smirnov (wKS) distance between the model
and the empirical distribution. B, The plane of wKS distance of(m, b)
the specified model and the empirical data for terrestrial mammals
(lighter values correspond to smaller wKS). The distinctive groove (line)
demonstrates that a systematic relationship between the bias parameter
m and the extinction parameter b is necessary to produce realistic mass
distributions (small wKS). The parameter pair used in A is marked.
Dashes demarcate the line of no within-lineage bias ( ). A colorm p 0
version of this figure is available in the online edition of the American
Naturalist.
roy’s (2008) ancestor-descendant data for North American
terrestrial mammals. Incorporating these values into equa-
tion (4) leaves only b unspecified. A strong test of this
model would estimate b from fossil data; however, while
studies of extinction among mammals suggest that b
1 0
(Liow et al. 2008), current data do not appear to be suf-
ficiently detailed to give a precise estimate for mammals.
Instead, following Clauset and Erwin (2008), we choose
b by minimizing the tail-weighted Kolmogorov-Smirnov
(wKS) goodness-of-fit statistic (Press et al. 1992) for the
predicted and empirical distributions:
FS(x) ⫺ P(x)F
wKS p max ,
冑
P(x)(1 ⫺ P(x))
x
where S(x) is the empirical distribution function and P(x)
is the predicted cumulative distribution function. Thus,
small values of wKS correspond to a model that is statis-
tically close to the empirical data everywhere. We find that
two alternative methods of choosing b, by numerically
matching the modal masses of the model and the empirical
data or by matching the expected maximum mass of the
model with the observed maximum in the empirical data,
produce similar results.
For the interested reader, details on the preparation of
the empirical data are discussed at length by Alroy (1998)
and Smith et al. (2003, 2007) for mammalian fauna and
by Dunning (2007) for avian fauna. In general, body mass
estimates were derived using conventional techniques (e.g.,
dentition techniques for mammals; Damuth 1990). For
simplicity, differences due to sexual dimorphism, geo-
graphic variation, and so on were ignored or averaged out.
Although such differences can be critical for smaller stud-
ies, given the scale of our data in terms of the number of
species studied and the wide range of body masses, mild
misestimates of body masses are unlikely to change our
conclusions unless they are widespread and systematic.
The resulting fit (fig. 2A) is in good agreement with the
empirical data, except for a slight overestimate of the num-
ber of species with mass near 1 kg, an underestimate of
the number near 300 kg, and a slight misestimate of the
number of very small-bodied species. The deviations in
the right tail are also seen in the CE model and may be
due to, for example, phylogenetically correlated speciation
or extinction events in the recent past. The deviations in
the left tail may be due to our omission of the mass de-
pendence in the drift term m identified by Clauset and
Erwin (2008); however, incorporating this behavior into
our diffusion-reaction model is technically nontrivial.
Thus, the evolution of mammalian species body masses
can largely be viewed as a simple diffusion process char-
acterized by (1) a slight within-lineage drift toward larger
masses over evolutionary time (Cope’s rule), (2) a hard
lower boundary on how small body masses can become,
and (3) a very soft constraint on large body masses in the
form of increased extinction risk. Phrased more concep-
tually, the left tail of the mammalian body mass distri-
260 The American Naturalist
Figure 3: A, Steady state solution of the diffusion-reaction model of
species body mass evolution (eq. [4]), with , , and bm p 0 M p 2g
min
chosen to minimize the tail-weighted Kolmogorov-Smirnov (wKS) dis-
tance between the model and the empirical distribution, with (smoothed)
empirical data for 8,617 avian species. B, The plane of wKS dis-(m, b)
tances (lighter values correspond to smaller wKS), showing a systematic
relationship between m and b for producing realistic (small wKS) avian
mass distributions. The form of this relationship is similar to that of
mammals (fig. 2B), suggesting a similar overall macroevolutionary mech-
anism. A color version of this figure is available in the online edition of
the American Naturalist.
bution is mainly controlled by the lower limit on mass,
while the right tail is the result of an evolutionary trade-
off at different timescales: over the short-term, within-
lineage increases in body mass offer selective advantages—
such as better tolerance of resource fluctuations, better
thermoregulation, better predator avoidance, and so on
(Calder 1984; Brown 1995)—while they also increase the
long-term risk of extinction, a trade-off previously iden-
tified in the more specific case of carnivorous mammals
(Van Valkenburgh 1999; Van Valkenburgh et al. 2004).
Avian Body Mass Evolution
Unlike mammals, data on most other taxonomic groups
are generally not sufficient to yield accurate estimates of
the parameters m and b (but see Novack-Gottshall and
Lanier 2008). The evolutionary history of mammalian
body masses is relatively clear, in part because mammalian
fossils are relatively plentiful and are often sufficiently well
preserved that body mass estimates can be made (Damuth
1990; Van Valkenburgh 1990) and because the distribution
of species body masses during an apparently stable evo-
lutionary period is known. Avian species, however, present
an interesting case for study using our model; the distri-
bution of extant avian body masses (fig. 3A) is relatively
well characterized (Dunning 2007), and evidence of a min-
imum species body mass M
min
is reasonable (Pearson
1950). However, the avian fossil record may be too sparse
to yield accurate estimates of m and b (Fountaine et al.
2005; Hone et al. 2008).
Even without estimates of m and b, however, the dif-
fusion model can be used to make quantitative statements
about the general character of avian body mass evolution.
To demonstrate this, we consider which combinations of
the parameters m, b produce “realistic” mass distributions,
that is, those with a small distributional distance to the
empirical distribution. In particular, we compute the wKS
distance between the model and the empirical data over
the plane and determine the regions that yield the(m, b)
best fits. To illustrate this technique in a better-understood
context, we first apply it to the data on Recent terrestrial
mammals, disregarding for the moment that we have an
estimate of m from fossil data.
The result of this exercise (fig. 2B) shows that realistic
mammalian mass distributions can be produced by a wide,
but not arbitrary, variety of biases m and extinction risks
b, including no bias at all; that is, . This degeneracy,m p 0
which forms a groove in the plane following roughly(m, b)
with , suggests that part of the difficulty in
v
m ∝ bv≈ 0.5
determining for a particular taxon whether mass evolution
is biased toward larger sizes (see, e.g., Maurer et al. 1992;
Maurer 1998; Bokma 2002) is that a positive bias is not
a necessary condition for the evolution of realistic mass
distributions. Indirect tests of the sign of m, based either
on the mass distribution within subclades (McShea 1994;
Wang 2001) or on changes to minimum and maximum
masses within a subclade over geologic time (Jablonski
1997), may be adequate if confounding hypotheses can be
eliminated or if an appropriate null model is available. On